The generalized Black Scholes Model refers to a stock dynamic that satisfy

$$ dS(t)=S(t)(\mu_t dt+ \sigma_t dW(t)) $$

By martingale representation theorem, it seems that if there is a risk neutral probability measure, then all stock dynamic is enclosed by the GBS.

Are there exceptions?

  • $\begingroup$ This model doesn’t include jumps in the asset price $\endgroup$ – Kevin Jul 21 '20 at 8:39
  • $\begingroup$ @KeSchn I think using, for example Dirac delta function in the $\mu_t$ can get around the theoretical stock price decrease during dividend payment $\endgroup$ – Preston Lui Jul 21 '20 at 8:49
  • $\begingroup$ Can you capture autocorrelation in $\sigma_tdW(t)$? $\endgroup$ – Bob Jansen Jul 21 '20 at 9:02
  • $\begingroup$ I was more thinking about something like Merton's (1976) or Bates' (1996) model where you add a Poisson process as additional risk source. So you have jumps at random times with random jump size. $\endgroup$ – Kevin Jul 21 '20 at 9:03
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    $\begingroup$ @BobJansen You're right, we would need $\mu_t=\mu(t,S_t)$ and $\sigma_t=\sigma(t,S_t)$. The latter would allow for the autocorrelation/stochastic volatility features you mentioned. If one includes a Dirac delta function in the mean though, that may affect integrability conditions and existence of solutions? The integral over $\mu_t$ would then only be $\mu_t$ evaluated at the mass point of the DD function. But the model certainly does not allow for standard jump processes such as Merton or Kou. $\endgroup$ – Kevin Jul 21 '20 at 10:24

KeSchn and I pointed out in the comments that this it is not possible to represent all stock dynamics using the Generalized Black Scholes model. For example, there can be jumps at random moments and not just at random moments but also jumps of random size. These jumps can affect either $\mu_t$ or $\sigma_t$. Models with too many sources of randomness are not considered useful but at least 2 extra source can be useful.

What does this say about the Martingale Representation Theorom. Doesn't that claim that stock dynamics can be captured using an Itô process? Unfortunately, the theorem is a bit more narrow (Wikipedia):

The martingale representation theorem states that a random variable that is measurable with respect to the filtration generated by a Brownian motion can be written in terms of an Itô integral with respect to this Brownian motion.

Emphasis mine. As long if there is one Brownian motion driving the randomness, all is good. The theorem doesn't hold with more sources than one.


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